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Describing periodic motion

The hour hand of the large clock on the wall in Union Station measures 24 inches in length. At noon, the tip of the hour hand is 30 inches from the ceiling. At 3 PM, the tip is 54 inches from the ceiling, and at 6 PM, 78 inches. At 9 PM, it is again 54 inches from the ceiling, and at midnight, the tip of the hour hand returns to its original position 30 inches from the ceiling. Let y equal the distance from the tip of the hour hand to the ceiling x hours after noon. Find the equation that models the motion of the clock and sketch the graph.

Begin by making a table of values as shown in [link] .

x y Points to plot
Noon 30 in ( 0 , 30 )
3 PM 54 in ( 3 , 54 )
6 PM 78 in ( 6 , 78 )
9 PM 54 in ( 9 , 54 )
Midnight 30 in ( 12 , 30 )

To model an equation, we first need to find the amplitude.

| A | = | 78 30 2 |      = 24

The clock’s cycle repeats every 12 hours. Thus,

B = 2 π 12     = π 6

The vertical shift is

D = 78 + 30 2     = 54

There is no horizontal shift, so C = 0. Since the function begins with the minimum value of y when x = 0 (as opposed to the maximum value), we will use the cosine function with the negative value for A . In the form y = A cos ( B x ± C ) + D , the equation is

y = −24 cos ( π 6 x ) + 54

See [link] .

Graph of the function y=-24cos(pi/6 x)+54 using the five key points: (0,30), (3,54), (6,78), (9,54), (12,30).
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Determining a model for tides

The height of the tide in a small beach town is measured along a seawall. Water levels oscillate between 7 feet at low tide and 15 feet at high tide. On a particular day, low tide occurred at 6 AM and high tide occurred at noon. Approximately every 12 hours, the cycle repeats. Find an equation to model the water levels.

As the water level varies from 7 ft to 15 ft, we can calculate the amplitude as

| A | = | ( 15 7 ) 2 |      = 4

The cycle repeats every 12 hours; therefore, B is

2 π 12 = π 6

There is a vertical translation of ( 15 + 8 ) 2 = 11.5. Since the value of the function is at a maximum at t = 0 , we will use the cosine function, with the positive value for A .

y = 4 cos ( π 6 ) t + 11

See [link] .

Graph of the function y=4cos(pi/6 t) + 11 from 0 to 12. The midline is y=11, three key points are (0,15), (6,7), and (12, 15).
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The daily temperature in the month of March in a certain city varies from a low of 24 °F to a high of 40 °F . Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point 32 °F . Let t = 0 correspond to noon.

y = 8 sin ( π 12 t ) + 32
The temperature reaches freezing at noon and at midnight.

Graph of the function y=8sin(pi/12 t) + 32 for temperature. The midline is at 32. The times when the temperature is at 32 are midnight and noon.
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Interpreting the periodic behavior equation

The average person’s blood pressure is modeled by the function f ( t ) = 20 sin ( 160 π t ) + 100 , where f ( t ) represents the blood pressure at time t , measured in minutes. Interpret the function in terms of period and frequency. Sketch the graph and find the blood pressure reading.

The period is given by

2 π ω = 2 π 160 π       = 1 80

In a blood pressure function, frequency represents the number of heart beats per minute. Frequency is the reciprocal of period and is given by

ω 2 π = 160 π 2 π = 80

See the graph in [link] .

Graph of the function f(t) = 20sin(160 * pi * t) + 100 for blood pressure. The midline is at 100.
The blood pressure reading on the graph is 120 80   ( maximum minimum ) .
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Modeling harmonic motion functions

Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.

Questions & Answers

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The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
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Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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